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Davenport-Hasse relations and an explicit Langlands correspondence. II: Twisting conjectures. (English) Zbl 1161.11409

Summary: Let \(F/\mathbb{Q}_p\) be a finite field extension. The Langlands correspondence gives a canonical bijection between the set \(\mathcal{G}^0_F (n)\) of equivalence classes of irreducible \(n\)-dimensional representations of the Weil group \(\mathcal{W}_F\) of \(F\) and the set \(\mathcal{A}^0_F (n)\) of equivalence classes of irreducible supercuspidal representations of \(\text{GL}_n(F)\). This paper is concerned with the case where \(n = p^m\). In earlier work, the authors constructed an explicit bijection \(\pi : \mathcal{G}^0_F (p^m) \rightarrow \mathcal{A}^0_F (p^m)\) using a non-Galois tame base change map. If this tame base change satisfies a certain conjectured automorphic Davenport-Hasse relation, and there exists a Langlands correspondence in \(p\)-power degree, then \(\pi \) is the Langlands correspondence. This paper is concerned with the problem of showing, without assuming a priori the existence of the Langlands correspondence, that (on the Davenport-Hasse conjecture) \(\pi \) preserves local constants of pairs, and so is a Langlands correspondence. The principal obstruction is the lack of knowledge of certain elementary properties of the local constant \(\varepsilon (\pi _1 \times \pi _2, s, \psi _F)\) for \(\pi _i \in \mathcal{A}^0_F (p^{m_i})\). We state these properties as conjectures (which are certainly true, as consequences of the existence of the Langlands correspondence and analogous properties of the Langlands-Deligne local constant) and show that they imply the desired result: \(\pi \) is a Langlands correspondence. In the process, we prove several new unconditional results concerning \(\pi \), and give a complete account of the rationality properties of \(L\)-functions and local constants of pairs for \(\text{GL}_n(F)\).
Part I, J. Reine Angew. Math. 519, 171–199 (2000; Zbl 1029.11063).

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 1029.11063
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References:

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